Accounting for Correlations When Fitting Extra Cosmological Parameters. (arXiv:1904.10521v1 [astro-ph.CO])

<a href="http://arxiv.org/find/astro-ph/1/au:+Huang_Y/0/1/0/all/0/1">Yajing Huang</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Addison_G/0/1/0/all/0/1">Graeme Addison</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Bennett_C/0/1/0/all/0/1">Charles Bennett</a>

Current cosmological tensions motivate investigating extensions to the

standard $Lambda$CDM model. Additional model parameters are typically varied

one or two at a time, in a series of separate tests. The purpose of this paper

is to highlight that information is lost by not also examining the correlations

between these additional parameters, which arise when their effects on model

predictions are similar, even if the parameters are not varied simultaneously.

We show how these correlations can be quantified with simulations and Markov

Chain Monte Carlo (MCMC) methods. As an example, we assume that $Lambda$CDM is

the true underlying model, and calculate the correlations expected between the

phenomenological lensing amplitude parameter, $A_L$, the running of the

spectral index, $n_{rm run}$, and the primordial helium mass fraction, $Y_P$,

when these parameters are varied one at a time along with the $Lambda$CDM

parameters in fits to the $textit{Planck}$ 2015 temperature power spectrum.

These correlations are not small, ranging from 0.31 ($A_L-n_{rm run}$) to

$-0.93$ ($n_{rm run}-Y_P$). We find that the values of these three parameters

from the $textit{Planck}$ data are consistent with $Lambda$CDM expectations

within $0.9sigma$ when the correlations are accounted for. This does not

explain the 1.8-2.7$sigma$ $textit{Planck}$ preference for $A_L>1$, but

provides an additional $Lambda$CDM consistency test. For example, if $A_L>1$

was a symptom of an underlying systematic error or some real but unknown

physical effect that also produced spurious correlations with $n_{rm run}$ or

$Y_P$ our test might have revealed this. We recommend that future cosmological

analyses examine correlations between additional model parameters in addition

to investigating them separately, one a time.

Current cosmological tensions motivate investigating extensions to the

standard $Lambda$CDM model. Additional model parameters are typically varied

one or two at a time, in a series of separate tests. The purpose of this paper

is to highlight that information is lost by not also examining the correlations

between these additional parameters, which arise when their effects on model

predictions are similar, even if the parameters are not varied simultaneously.

We show how these correlations can be quantified with simulations and Markov

Chain Monte Carlo (MCMC) methods. As an example, we assume that $Lambda$CDM is

the true underlying model, and calculate the correlations expected between the

phenomenological lensing amplitude parameter, $A_L$, the running of the

spectral index, $n_{rm run}$, and the primordial helium mass fraction, $Y_P$,

when these parameters are varied one at a time along with the $Lambda$CDM

parameters in fits to the $textit{Planck}$ 2015 temperature power spectrum.

These correlations are not small, ranging from 0.31 ($A_L-n_{rm run}$) to

$-0.93$ ($n_{rm run}-Y_P$). We find that the values of these three parameters

from the $textit{Planck}$ data are consistent with $Lambda$CDM expectations

within $0.9sigma$ when the correlations are accounted for. This does not

explain the 1.8-2.7$sigma$ $textit{Planck}$ preference for $A_L>1$, but

provides an additional $Lambda$CDM consistency test. For example, if $A_L>1$

was a symptom of an underlying systematic error or some real but unknown

physical effect that also produced spurious correlations with $n_{rm run}$ or

$Y_P$ our test might have revealed this. We recommend that future cosmological

analyses examine correlations between additional model parameters in addition

to investigating them separately, one a time.

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